“We want to look for ways to find common ground to get a budget agreement,” Ryan, the chairman of the House Budget Committee, said after a breakfast with Murray and other budget leaders.

“Our goal to do good for the American people, to get this debt under control, to do smart deficit reduction, and to do things that we think can grow the economy and get people back to work," said Ryan, who voted against the deal to raise the debt ceiling and end the shutdown. "Those are our shared goals.”

Murray said the breakfast held just off the Senate floor was a “good conversation.”

“We believe there is common ground, and we can show the American people that Congress can work,” Murray said.

The budget conference faces a Dec. 13 deadline for reporting back to Congress. Their goal is to produce a final budget top-line number for the current fiscal year so that final spending bills can be worked out before funding runs out again on Jan. 15.

Analysts are already giving Ryan and Murray slim chances for success. The new budget conference comes after the failure of a 2011 supercommittee, and after the Simpson-Bowles commission failed in 2010 to get enough support to force votes on a $4 trillion deficit grand bargain.

Murray said the new conference has much more “limited” goals than the supercommittee. Its prime focus will be on the sequester for the current fiscal year and finding either entitlement or revenue replacements for those cuts.

“We have a challenge that has been handed to us to have a reconciliation between the House budget and the Senate budget,” she said.

Murray and Ryan dodged when asked how they would reconcile Democratic demands for more revenue and Republican insistence so far on deficit reduction without tax increases.

“It’s premature exactly how we are going to do that,” Ryan said when asked about the details going forward.

While the GOP took the biggest beating in the polls during the 16-day government shutdown, Democrats also saw their popularity shrink. That could give both sides incentive to reach a deal.

There already has been talk about a deal involving trading off the sequester spending cuts for changes to entitlement programs, which Republicans see as the drivers of the deficit. But Democrats are expected to press that a deal also include new taxes, something Republicans say they would not accept.

Van Hollen said that he hopes the morning’s meetings signals brinksmanship over the debt ceiling and government shutdowns are over.

“I hope that one lesson that came out of this totally unnecessary shutdown is that nobody should try to advance a partisan agenda by trying to shut down the government,” he said. “I hope people put away the clubs and come to the table for a traditional negotiation.”

The full budget conference is set to have its first formal meeting the week of Oct. 28 when both the House and the Senate return to work.

“Chairman Murray is very knowledgeable about these issues and a strong leader,” said Sessions. “There are a number of things I know we can agree on. I don’t want to raise expectations above reality. … We may be able to accomplish something good.”